The key to any fighting game is to learn the weapons that are provided to you. In Saint's Row, make sure that you match your weapon to the level you're on. For example if you're going into a gang fight and everybody there has a certain gun, carry that gun too. That way when they all drop their ammo, you can use it. You can only carry one type of ammo on you.

If you're fighting a cop, they will match your weapon. So if you go in with fists, they won't pull their gun. However if you shoot them with a gun, they will shoot back. If you kill them, they only drop what they were using at the time.

Note that when you buy a weapon, it comes with two magazines worth of ammo.

Pistols
For pistols, you can carry about 250 rounds total.

Vice 9 - 12-round clip - $100
Most bad guys carry a Vice 9 at first, so this is great for replacement ammo.

NR4 - 12-round clip - $400
This is what cops tend to carry, so this is great for cop fight situations.

.44 Shepherd - 6-round - $1,200
This is a large revolver, like a Magnum. Pretty good with one-shot kills.

GDHC .50 - 8-round clip - $1,000
This is about the same as the revolver as far as damage

SMG - Sub Machine Guns
SMGs are a bit more expensive than pistols and in the early missions you don't tend to get "free ammo" for them. You can carry about 650 rounds for these.

T3K Urban - 50-round clip - $500

SKR-7 Spree - 50-round clip - $650

Shotgun
Shotguns are the classic "fire tons of little balls" guns that people go duck hunting with in the woods. The can cause huge damage up close, but their range is quite limited. You can carry about 108 rounds max.